Find out if the following pairs of groups are isomorphic:
1 $(\mathbb {Z}_{7}\setminus\{0\}; .)$ with $(\mathbb {Z}_{6}; +)$
2 $(\mathbb {R}_{+}; .)$ with $(\mathbb {R}\setminus\{0\}; .)$
3 $(\mathbb {R}\setminus\{0\}; .)$ with $(\mathbb {C}\setminus\{0\}; .)$
4 $(\mathbb {R}; +)$×$(\mathbb {R}; +)$ with $(\mathbb {C}; +)$
5 $(\mathbb {Z}_{2}; +)$×$(\mathbb {Z}_{2}; +)$ with $(\mathbb {Z}_{4}; +)$
6 the group of symmetries of the equilateral triangle with group of the all permutations of {$1,2,3$}.
I know that I have to find some bijective function which is homomorphism. But it takes very long time to confirme all possibilities. I would like to now if there is any general way how to find out if two groups are isomorphic.
Thank you very much for any help.
Best Answer
The groups from item 1 are isomorphic because they are both cyclic groups of order $6$ ($(\mathbb{Z}_7\setminus\{0\},.)$ is generated by $3$).
The groups of item 2 are not isomorphic, because $(\mathbb{R}_+,.)$ has no element of order $2$, whereas $(\mathbb{R}\setminus\{0\},.)$ has such an element ($-1$).
In general, examining the orders of the elements is a good strategy to prove that two groups are not isomorphic.
Can you take it from here?